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To solve this problem, let’s go through each part step by step.

### Step 1: Find the Mean of the Sample Data
The temperatures measured are:
\[ 77.5, 74.9, 67.3, 83.9, 72.2, 86.7, 80.4, 71, 72.9, 86.7, 73.6, 86.8 \]

Let's calculate the sample mean \( \bar{x} \):
\[
\bar{x} = \frac{77.5 + 74.9 + 67.3 + 83.9 + 72.2 + 86.7 + 80.4 + 71 + 72.9 + 86.7 + 73.6 + 86.8}{12}
\]

### Step 2: Construct a 99% Confidence Interval for the Mean
1. The sample size \( n = 12 \).
2. The population standard deviation \( \sigma = 6.88 \).
3. For a 99% confidence level, the z-score (critical value) \( z_{\alpha/2} \) is approximately 2.576.

The formula for a confidence interval is:
\[
\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}
\]

### Step 3: Calculate the 95% Single-Sided Lower Bound
For a single-sided lower bound with 95% confidence, we use a z-score of approximately 1.645.

The lower bound is:
\[
\bar{x} - z \frac{\sigma}{\sqrt{n}}
\]

### Step 4: Determine the Sample Size Needed for a Desired Margin of Error
We need to find the required sample size \( n \) to have a margin of error \( E < 1 \) with 95% confidence. The formula for sample size with a given margin of error is:
\[
n = \left( \frac{z \sigma}{E} \right)^2
\]
where \( E = 1 \) and \( z = 1.645 \).

Let me calculate each of these steps.Here are the results for each part:

1. **Sample Mean** (\( \bar{x} \)): **77.825**

2. **99% Confidence Interval for the Mean**:
   \[
   72.71 \leq \mu \leq 82.94
   \]

3. **95% Single-Sided Lower Bound for the Mean**:
   \[
   \mu \geq 74.56
   \]

4. **Required Sample Size for Margin of Error \( E < 1 \) with 95% Confidence**:
   \[
   n = 129
   \]

If you need more details or have questions, let me know!

### Related Questions
1. What does a 99% confidence interval indicate about the mean?
2. How is the margin of error affected by the sample size?
3. Why is a larger sample size required for a smaller margin of error?
4. How would the confidence interval change if the standard deviation were different?
5. What if we wanted a 90% confidence interval—how would the z-score change?

### Tip
Remember, confidence intervals provide a range within which we expect the true population parameter to lie, based on the sample data and a specified confidence level.

An experiment yielded n = 12 temperature measurements, assumed to have a normal distribution with known standard deviation σ = 6.88. Calculate:
(a) The 99% two-sided confidence interval on the mean temperature.
(b) The 95% confidence single-sided lower bound on the mean.
(c) For 95% confidence that the estimated mean error E < 1, determine the required sample size.

Explore the calculation of a 99% confidence interval, 95% lower bound, and required sample size for temperature measurements. Given a sample of 12 observations with standard deviation 6.88, this problem uses statistical concepts of confidence intervals, sample mean, and margin of error. Ideal for undergraduate students in statistics.

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